Ann. Inst. Fourier, Grenoble 23, 2 (1973), 87-91.
EQUIVARIANT ALGEBRAIC TOPOLOGY
by Soren ILLMAN
This talk covered the main parts of my thesis [5], see also [4] and [6]. Our main objective is to provide the equivariant analogue of ordinary singular homology and cohomology, and the equivariant analogue of the theory of CW com- plexes. For actions of discrete groups we have the work by Bre- don [I], [2], and more recently also the article by Brocker [3].
1. Equivariant singular homology and cohomology.
Let G be a topological group. By a G-space X we mean a topological space X together with a left action of G on X.
A G-pair (X, A) consists of a G-space X and a G-subspace A of X. The notions G-map, G-homotopy, G-homotopy equivalence/etc... have the usual meaning.
DEFINITION 1.1. — An orbit type family y for G is a family of subgroups of G which is closed under conjugation.
Thus the family of all subgroups and the family of all finite subgroups of G are examples of orbit type families for G.
A more special example is the following. Let G == 0(n) and let y be the family of all subgroups conjugate to 0(r) (standard imbedding) for some r, where 0 ^ r ^ n.
Let R be a ring with unit. By an R-module we mean a unitary left R-module.
DEFINITION 1.2. —Let' 3F be an orbit type family for G.
A coyariant coefficient system k for ^", o^er the ring R,
88 SOBEN I L L M A N
is a covariant functor from the category of G-spaces of the form G/H, where H e ^, and G-homotopy classes of G-maps^
to the category of R-modules.
A contravariant coefficient system m is defined by the contravariant version of the above definition.
THEOREM 1.1. — Let G be a topological group. Let y be an orbit type family for G, and k a covariant coefficient system for ^, over the ring R. Then there exists an equiva- riant homology theory H^( ; k) defined on the category of all G-pairs and G-maps (and with values in the category of
^{-modules), which satisfies all seven equivariant Eilenberg"
Steenrod axioms, and which has the given coefficient system k as coefficients.
By the statement that H^( ; k) satisfies the equivariant dimension axiom and has k as coefficients we mean the following. If H e ^ then
H^(G/H;/c) = 0 for n ^ 0, and there exists a natural isomorphism
Y : Ho^G/H^-^/cCG/H).
The meaning of the rest of Theorem 1.1. is clear.
THEOREM 1.2. — Let G and, ^ be as above, and let m be a contravariant coefficient system for ^\ over the ring R.
Then there exists an equivariant cohomology theory HS( ; m) defined on the category of all G-pairs and G-maps (and with values in the category of I{-modules), which satisfies all seven equivariant Eilenberg-Steenrod axioms^ and which has the given coefficient system m as coefficients.
For the details we refer to [5] and [6]. We call H^( ; k) for equivariant singular homology with coefficients in /c, and Ht( ; m) for equivariant singular cohomology with coefficients in m. Further properties, like functoriality in the transformation group G, transfer homomorphisms, Kronecker index, and a cup-product for equivariant singular cohomology are also given in [5] and [6].
EQUIVARIANT ALGEBRAIC TOPOLOGY g9
2. Eqmvariant CW complexes/
In this section G denotes a compact Lie group.
DEFINITION 2.1. — Let X beaHausdorff G-space and A a closed G-subset of X, and n a non-negative integer. We say that X is obtainable from A by adjoining equivariant n-cells if there exists a collection {c;},^ of closed G'subsets of X such that.
1) X = A u (\^) (^ \ and X has the topology coherent with {A, c;}^j.
2) Denote c] = c} n A, ^n
(c;-^) n ( c ? ~ c ? ) = 0 if ^/.
3) For each j e j there exists a closed sub group H, of G and a G-map
fj: {En x G/H,, S^ x G/H,) -^ (c;, c})
^cA^at /.(E" x G/H,) = c}, anrf /) maps (E" - S"-1) X G/H, homeamorphically onto c^ — c^
DEFINITION 2.2. — An equwariant relative CW complex (X, A) consists of a Hausdorff G'space X, a closed G-subset A o/1 X, an^ an increasing filtration of X by closed G'subsets (X, A)* k =s 0 , 1 , ..., such that.
1) (X, A)0 i5 obtainable from A by adjoining equivariant O'cells, and for k ^ 1(X, A)* is obtainable from (X, A)^1
by adjoining equivariant k-cells.
2) X == ^_J (X, A)^, an^ X Aa^ ^e topology coherent with {(X^A)^.
The closed G-subset (X, A)* is called the k-skeleton of (X, A). If A == 0 we call X an equivariant CW complex and denote the k-skeleton by X\
It is not difficult to prove the equivariant analogues of the standard elementary properties of CW -complexes that is,
90 SOBEl^ ILLMAN
the equivariant homotopy extension property for an equi- variant relative CW complex (X. A), the equivariant skeletal approximation theorem, and an equivariant White- head theorem. These results are stated as Propositions 2.3- 2.5 in [4], and will not be repeated here. I understand that both C. Vaseekaran and S. Willson have also proved these three results.
Our main result about equivariant CW complexes is the following theorem, see [4] and [5].
THEOREM 2.1. — Every differentiable G-manifold M is an equivariant CW complex.
This result has also been proved by T. Matsumoto, see [7]
Proposition 4.4. In fact a stronger result is true. We proved in [5] that every differentiable G-manifold can be equiva- riantly triangulated. The proof of this uses the theorem of C.T. Yang [9] that the orbit space G/M can be triangulated, the existence of slices, and the « covering homotopy theorem » of Palais [8] Theorem 2.4.1. It should be observed that Pro- position 4.4. in Matsumoto [7] also proves this stronger result.
3. Equivariant singular homology and cohomology of finite dimensional equivariant CW complexes.
We are still assuming that G is a compact Lie group. An equivariant CW complex X is called finite dimensional if X == X"1 for some m. Let us for simplicity assume that the orbit type family y under consideration is the family of all closed subgroups of G. We say that a covariant coefficient system k is finitely generated if /c(G/H) is a finitely gene- rated R-module for every closed subgroup H of G.
THEOREM 3.1. — Let X be a finite dimensional equiva- riant CW complex. Then the n-th homology of the chain complex
... ^H^X^ X"-2; /c^H^X", X71-1; /c)<l ...
is isomorphic to H^(X; k).
Together with Theorem 2.1 this gives us.
EQUIVABIANT ALGEBRAIC TOPOLOGY 91
COROLLARY 3.2. - Let M"1 be an m-dimensional differen' liable G-manifold. Then
HWm;k)=0 for r > m .
If M^ moreover is compact, and k is a finitely generated coefficient system over a noetherian ring R, then H^M"*; k) is a finitely generated R-module for every r.
The corresponding results for cohomology are true.
BIBLIOGRAPHY
[1] G. BREDON, Equivariant cohomology theories, Bull. Amer. Math Soc 73 (1967), 269-273. ' 7 [2] G. BREDON, Equivariant cohomology theories, Lecture Notes in Mathe-
matics, Vol. 34, Springer-Verlag (1967).
[3] T. BROCKER, Singulare Definition der Aquivarianten Bredon Homologie Manuscripta Matematica 5 (1971), 91-102. ? [4] S. ILLMAN, Equivariant singular homology and cohomology for actions of compact Lie groups. To appear in: Proceedings of the Conference on Transformation Groups at the University of Massachusetts, Amherst, June 7-18 (1971) Springer-Verlag, Lecture Notes in Mathematics.
[5J S. ILLMAN, Equivariant Algebraic Topology, Thesis, Princeton Uni- versity (1972).
[6] S. ILLMAN, Equivariant singular homology and cohomology To appear in Bull. Amer. Math. Soc.
[7] T. MATSUMOTO, Equivariant K-theory and Fredholm operators, Journal of the Faculty of Science, The University of Tokyo, Vol. 18 (1971),
luy-iAo.
[8] R. PALAIS, The classification of G-spaces, Memoirs of Amer. Math Soc 36 (1960).
[9] C. T. YANG, The triangulahility of the orbit space of a differentiable transformation group, Bull. Amer. Math. Soc., 69 (1963), 405-408.
Soren ILLMAN, Princeton University and University of Helsinki.
